If the integral from a to b of f(x)dx=a+2b then integral from a to b of (f(x)+3)dx= ?
The answer: 5b-2a
How do I get to that answer??
integral of (f(x) + 3)dx = integral of f(x)dx + integral of 3dx
integral from a to b of (f(x)+3)dx
= integral from a to b of f(x)dx + 3(b-a)
= a + 2b +3b -3a
= 5b - 2a
Well, dear human, let's juggle some mathematical fun here to find the answer!
First, we start with the integral from a to b of f(x)dx, which we know equals a + 2b. Now, if we add 3 to the function f(x), we get f(x) + 3.
So, we want to find the integral from a to b of (f(x) + 3)dx. To do that, we simply integrate f(x) + 3 over the bounds from a to b.
Now, the integral of f(x) + 3 with respect to x is just the sum of the integrals of f(x) and 3 separately.
Since the integral of 3dx is just 3x, integrating f(x) + 3 from a to b would give us the integral of f(x) over the same bounds (a to b) plus the integral of 3x over the same bounds.
The integral of f(x) over the interval from a to b is a + 2b and the integral of 3x over the same interval is 3(b - a).
So, to find the integral from a to b of (f(x) + 3)dx, we add these two integrals together.
(a + 2b) + (3(b - a)) simplifies to a + 2b + 3b - 3a, which further simplifies to 5b - 2a.
And there you have it! The answer is 5b - 2a. Ta-da!
To find the integral of a function, you can use the fundamental theorem of calculus. This theorem states that if F(x) is an antiderivative of f(x) on an interval [a, b], then the definite integral of f(x) from a to b is equal to F(b) - F(a).
Let's apply this theorem to solve the given problem step by step.
1. Start by finding the antiderivative of f(x):
∫f(x)dx = F(x) + C
2. Use the given information:
∫f(x)dx = a + 2b
3. Apply the fundamental theorem of calculus:
F(b) - F(a) = a + 2b
4. Now, we need to find F(b) and F(a).
F(b) = a + 2b + C
F(a) = C (since the constant of integration, C, cancels out when subtracting F(a) from F(b))
5. Now substitute F(b) and F(a) back into the equation from step 3:
(a + 2b + C) - C = a + 2b
6. Simplify the equation:
2b - a = a + 2b
7. Rearrange the terms:
-a = a
8. Multiply both sides by -1:
a = -a
9. Divide both sides by a (assuming a is not zero):
1 = -1
Since we arrive at a contradiction, this means that our initial assumption that F(x) exists and satisfies the given conditions is incorrect. Therefore, the given information is not consistent, and the problem cannot be solved as stated.
Hence, the integral from a to b of (f(x) + 3)dx cannot be determined based on the given information.